{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:CV2BHHORZSTODXREWQLBWEK6HF","short_pith_number":"pith:CV2BHHOR","canonical_record":{"source":{"id":"1606.05904","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-06-19T19:32:46Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"a058ed797b47d8a1f9617878af5cb40323feb09c49757ca6caa8322f2e0958bd","abstract_canon_sha256":"b8351c6c79b9df1e6c72a7adc5e475233eed22fe169bc618f8d592974ed0bed6"},"schema_version":"1.0"},"canonical_sha256":"1574139dd1cca6e1de24b4161b115e3958d0d0a7fbc9862b4ffa4d6dadb2d5f2","source":{"kind":"arxiv","id":"1606.05904","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.05904","created_at":"2026-05-18T01:12:15Z"},{"alias_kind":"arxiv_version","alias_value":"1606.05904v1","created_at":"2026-05-18T01:12:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05904","created_at":"2026-05-18T01:12:15Z"},{"alias_kind":"pith_short_12","alias_value":"CV2BHHORZSTO","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CV2BHHORZSTODXRE","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CV2BHHOR","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:CV2BHHORZSTODXREWQLBWEK6HF","target":"record","payload":{"canonical_record":{"source":{"id":"1606.05904","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-06-19T19:32:46Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"a058ed797b47d8a1f9617878af5cb40323feb09c49757ca6caa8322f2e0958bd","abstract_canon_sha256":"b8351c6c79b9df1e6c72a7adc5e475233eed22fe169bc618f8d592974ed0bed6"},"schema_version":"1.0"},"canonical_sha256":"1574139dd1cca6e1de24b4161b115e3958d0d0a7fbc9862b4ffa4d6dadb2d5f2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:15.848177Z","signature_b64":"y2wncKl9LrkvnSMW6B0bESHVnMCKDZey13pn6r9mSyMsliSowk4I6mDQGueg1sml4GnCWJ9zN+XEMzEa05P4CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1574139dd1cca6e1de24b4161b115e3958d0d0a7fbc9862b4ffa4d6dadb2d5f2","last_reissued_at":"2026-05-18T01:12:15.847630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:15.847630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.05904","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LPdnpRVMP09+Xbs2NYxnpXWnWYHHRocB36SmiUo4E1Hj3/12G9q2vQGkqmkssmzENdLk0pNtYY8RHTOgh9iODg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T11:51:53.198379Z"},"content_sha256":"411619523369fbb91f8c349fe0d4cc6652aa41c170439db7f3b5b11c70d5ce09","schema_version":"1.0","event_id":"sha256:411619523369fbb91f8c349fe0d4cc6652aa41c170439db7f3b5b11c70d5ce09"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:CV2BHHORZSTODXREWQLBWEK6HF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Message Dimensions of Vector Linearly Solvable Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Brijesh Kumar Rai, Niladri Das","submitted_at":"2016-06-19T19:32:46Z","abstract_excerpt":"It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is $2$ [3]. It is not known whether this result can be generalized for an arbitrary message dimension. In this paper, we show that there exists a network which admits an $m$ dimensional vector linear solution, where $m$ is a positive integer greater than or equal to $2$, but does not have a vector linear solution over any finite field when the message dimension is less than $m$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05904","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uIGEHXW25OuNE0YWTyzSHxd3HwAOm9/CvHzXl6BgHsi6WnCKUJdeEbsti9xkD+545vlzRp9FCjdkNsxSrbyeCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T11:51:53.198719Z"},"content_sha256":"643e29832f29e05c69f8c0a467b3f60a7897b03a6f4831770c0bf77d09eb382d","schema_version":"1.0","event_id":"sha256:643e29832f29e05c69f8c0a467b3f60a7897b03a6f4831770c0bf77d09eb382d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CV2BHHORZSTODXREWQLBWEK6HF/bundle.json","state_url":"https://pith.science/pith/CV2BHHORZSTODXREWQLBWEK6HF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CV2BHHORZSTODXREWQLBWEK6HF/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-30T11:51:53Z","links":{"resolver":"https://pith.science/pith/CV2BHHORZSTODXREWQLBWEK6HF","bundle":"https://pith.science/pith/CV2BHHORZSTODXREWQLBWEK6HF/bundle.json","state":"https://pith.science/pith/CV2BHHORZSTODXREWQLBWEK6HF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CV2BHHORZSTODXREWQLBWEK6HF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CV2BHHORZSTODXREWQLBWEK6HF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b8351c6c79b9df1e6c72a7adc5e475233eed22fe169bc618f8d592974ed0bed6","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-06-19T19:32:46Z","title_canon_sha256":"a058ed797b47d8a1f9617878af5cb40323feb09c49757ca6caa8322f2e0958bd"},"schema_version":"1.0","source":{"id":"1606.05904","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.05904","created_at":"2026-05-18T01:12:15Z"},{"alias_kind":"arxiv_version","alias_value":"1606.05904v1","created_at":"2026-05-18T01:12:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05904","created_at":"2026-05-18T01:12:15Z"},{"alias_kind":"pith_short_12","alias_value":"CV2BHHORZSTO","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CV2BHHORZSTODXRE","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CV2BHHOR","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:643e29832f29e05c69f8c0a467b3f60a7897b03a6f4831770c0bf77d09eb382d","target":"graph","created_at":"2026-05-18T01:12:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is $2$ [3]. It is not known whether this result can be generalized for an arbitrary message dimension. In this paper, we show that there exists a network which admits an $m$ dimensional vector linear solution, where $m$ is a positive integer greater than or equal to $2$, but does not have a vector linear solution over any finite field when the message dimension is less than $m$.","authors_text":"Brijesh Kumar Rai, Niladri Das","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-06-19T19:32:46Z","title":"On the Message Dimensions of Vector Linearly Solvable Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05904","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:411619523369fbb91f8c349fe0d4cc6652aa41c170439db7f3b5b11c70d5ce09","target":"record","created_at":"2026-05-18T01:12:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b8351c6c79b9df1e6c72a7adc5e475233eed22fe169bc618f8d592974ed0bed6","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-06-19T19:32:46Z","title_canon_sha256":"a058ed797b47d8a1f9617878af5cb40323feb09c49757ca6caa8322f2e0958bd"},"schema_version":"1.0","source":{"id":"1606.05904","kind":"arxiv","version":1}},"canonical_sha256":"1574139dd1cca6e1de24b4161b115e3958d0d0a7fbc9862b4ffa4d6dadb2d5f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1574139dd1cca6e1de24b4161b115e3958d0d0a7fbc9862b4ffa4d6dadb2d5f2","first_computed_at":"2026-05-18T01:12:15.847630Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:15.847630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"y2wncKl9LrkvnSMW6B0bESHVnMCKDZey13pn6r9mSyMsliSowk4I6mDQGueg1sml4GnCWJ9zN+XEMzEa05P4CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:15.848177Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.05904","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:411619523369fbb91f8c349fe0d4cc6652aa41c170439db7f3b5b11c70d5ce09","sha256:643e29832f29e05c69f8c0a467b3f60a7897b03a6f4831770c0bf77d09eb382d"],"state_sha256":"b9e1c2ca04d01bb1582b9a192ab7b8f386866f64b7fd495c15a55b7e18a21919"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cT0c/c3pCJByeg7WiaJdY2l77M/QezUrA7LKCgwTlhFi24q7IQJKjAbvIHBofqsMDAzevaSlJerGGAsKyYPeAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T11:51:53.200795Z","bundle_sha256":"f501677ed48495f4937d337fda8a2ebc4eaca460721e2d94aca3b9553fcc647a"}}